206 Circular Appearance of the Ellipse. (Marcu, 
Definition.—The straight line joining the vertex of the cone and 
the centre of the ellipse (which forms the base) is the axis of the 
cone. ‘The only case which bears upon the principal object of the 
present investigation is when this axis is perpendicular to the base; 
to this case, therefore, the first, s¢cond, and third, propositions are 
confined; the second, indeed, would be generally true, whatever 
inclination were given to the axis; but it is not my intention at 
present to go into the full consideration of the sections of the 
elliptic cone. 
Prop. 1.—Any plane passing through the axis will be perpendi- 
cular to the base, and its common section with the conical super- 
_ ficies will be an isosceles triangle. 
The first part of the proposition will be true by Eucl. (18, xi.) 
To prove the second part, let V C (Fig. 3) be the axis of the ellip- 
tical cone V A CB, and C the centre of the elliptical base. Then 
it may be shown exactly, as in the circular cone, that the common 
sections of the plane and conical superficies will be the straight 
lines VA, V B; consequently V AB isa triangle. Now asAB is 
a diameter of the elliptical base, and C is the centre, AC = CB. 
Hence A C? + VC? = C B* + V C*; but by (47, i.) AC? + 
VC? = V A’, and C B? + VC? = VB’; therefore V A? = 
V B.—Q. E. D. 
Prop. 2.—\f the conical superficies be cut by a plane parallel to 
the base, the common section will be an ellipse similar to the base. 
Let (in Fig. 4) DH EF be the common intersection of the 
conical superficies, and of the plane parallel to the base; then 
D H E is an ellipse similar to the base AGB. 
Let a plane pass through the axis, and likewise through any 
diameter, A B, of the base. Let its common section with the 
plane D H E be D E, and let the axis meet the plane DH EF in 
F. Then (16, xi.) D Eis parallel to AB, and DT: AC:: VF: 
VC::FE:C B. Hence DF: AC::FE:CB, andAC= 
CB; therefore DF = FE. Now this will be true, whatever 
may be the position of A B and D E; therefore any line terminated 
by D H E, and passing through F, will be bisected in that point. 
Again, let GC be a semidiameter conjugate to A C, and through 
GC, VC passa plane, and let its common section with the plane 
DEHFbe HF. Then, as before, H F is parallel to G C, and 
DF:AC:: HF: CG; therefore alternately DF: HF :: AC: 
C G, and (10, xi.) the angle D F H is equal to the angle ACG; 
and as this will be true of any conjugate diameters, it follows that 
DH Eis an ellipse similar to A G B, and that its centre is at F. 
Cor.—D F*: F H® :: AC*: CG*. Hence the rectangle under 
the abscissee of D E : the square of its ordinate :; A C?: C G*. 
Prop. 3.—Let V AB G be a cone with an elliptic base, of which 
_ G Ois the greater, and A B the smaller, axis ; and let the common 
section of a plane passing through the axis of the cone and the 
minor axis, A B, of the ellipse, be the isosceles triangle V A B: 
in A B produced take the point T, such that O G* — A B®: O G 
:: VB*: V T®. Then by lemma 2, A B*: OG*:: AT, TB: VT*% 
